'N7 7 DA4 5 AU) WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER NDA4 5 DIRECT APPROACH TO THE VILLARCEAU CIRCLES OF AJORUS /

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1 NDA4 5 DIRECT APPROACH TO THE VILLARCEAU CIRCLES OF AJORUS / 7 DA4 5 AU) WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER I J SCHOENBERG MAR 84 MRC-TSR-2651 DAAG29 80 C 0041 UNCLASSIFIED /121 N 'N7

2 Ifl=j~.2 2 MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS A

3 in MRC Technical Summary Report #2651 A DIRECT APPROACH TO THE VILLARCEAU CIRCLES OF A TORUS I. J. Schoenberg Mathematics Research Center University of Wisconsin-Madison 610 Walnut Street Madison, Wisconsin March 19R~4 uj (Received December 16, 1983) C.2 Sponsored by Approved for public release Distribution unlimited S D I ELECTE f U. S. Army Research Office E P. 0. Box Research Triangle Park Worth Carolina ~ 5 3 8

4 UNIVERSITY OF WISCONSIN-MADISON MATHEMATICS RESEARCH CENTER A DIRECT APPROACH TO THE VILLARCEAU CIRCLES OF A TORUS I. J. Schoenberg Technical Summary Report #2651 March 1984 ABSTRACT Let T(r,c) denote a torus enveloped by a sphere of radius c whose center describes a circle r of radius a, (c < a). We call r the central circle of the torus T(r,c). Let T(r,c 1 ) and T(F',c) denote two tori satisfying the following conditions: 1. The central circles r and r' have equal radii, both - a. 2. The two tari T(r,c 1 ) and T(rI,c;) are linked, like the two consecutive elements of a chain. This requires that a > c + c;. It is shown that the two tori T(r,c I ) and T(r',c;) can be placed in such a position that they are tangent to each other along a closed curve y (which is not a circle) without any gaps between the tori. The simple closed curve y is not a circle., 'it is also shown that this property is equivalent to the circles on the torus discovered by the astronomer Yvon Villarceau in The paper is to appear in the Dutch periodical "Simon Stevin". I ANS(MOS) Subject Classification: 51-01, 01-A55 Key Words: The torus, Helicoidal motion of a circle. Work Unit Number 6 - Miscellaneous Topics 1/ Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

5 SIGNIFICANCE AND EXPLANATION Let T and T' be two tori which are linked like the two consecutive elements of a chain. Moreover-wf assumesthat T and TO have qentral c rcles of equal radii. By central circle of a torus we mean the locus of the center of the sphere of constant radius which envelopes the torus. It is shown that the linked tori can be so placed that they are tangent to each other along simple closed curve 3 which is not a circle. In this mutually tangent position there is no gap between the two tori T and T'. It is shown that the above property is equivalent to the (slanting) circles of a torus discovered by Yvon Villarceau in Accezilntn For NTT ' I- ~ j Dint ll.vlbilit-r Codes Avaii and/or Special!I t The responsibility for the wording and views expressed in this descriptive smiary lies with MRC, and not with the author of this report.

6 A DIRECT APPROACH TO THE VILLARCEAU CIRCLES OF A TORUS I. J. Schoenberg A well known result states that if we cut a horizontal torus T by a slanting plane which is tangent in two (hyperbolic) points of the surface of T, we obtain as section two equal circles called the Villarceau circles of T*. A direct approach to these circles is described in the present note. Fig. 1 shows the horizontal circle r - ABC, of center 0 and radius a, where AOC is a diameter and LAOB We are also given c such that (1) a > c > 0, and we define b > 0 by (2) b 2 + c 2 = a 2. On OC we mark the point 0' so that 00' - c, and draw O'D" 1 O'O such that D e r. Evidently, by (2) we have O'D b. At D we erect the vertical line DE and choose E so that DE - c. From (2) we conclude that O'I - a. We now perform in succession on r the following two rigid motions: 1. We translate F along OC by the amount c, which moves 0 to 0', and B to F, where O'F - a. Call F the new position of F. 2. We rotate the circle F around AC by the angle LFO'E -a, which moves F to Z. From the triangle O'DE we see that ()b c (3) Cos a "'k a sinai-. a S lee (11, (2, pp and [3]. Sponsored by the United States Army under Contract No. DAAG29-80-C J //

7 z z I I ' Figure 1. I We denote by F' = A'EC' the final position of bi F. Notice that by our construction the three segments AA', DE, CC' satisfy I ~AA' DE CC' =c and that they are common normals of F and r'. It seems remarkable that all X,$ x, cosmion normals of F and F' are of constant length =c. This property and.1 further details of the relationship between F and F' are described by the i ' following theorem., Theorem 1. On F and F' we select points P and P', respectively, such that (4) 8 "LAOP, * =LAO'P', (0 ( 8, * < 2), ~-2- i4...

8 and we use these notations throughout. There is a 1- mppn (5) v: P + P,, (P e r, P, e r'), of r onto r* defined by the equations (a co coos8+c sn b sino0 a+c cos 'ac o the inverse mappin V1being given by (7) coso a aco c sie bsin # a-c coo *' i a -c coo* The properties of PI- VP are as follows: (8) PP' - c for all P e r. (9) The vector Ptis perpendicular to the tangents at P and P' of r and r,, respectively. (10) If p e r, p, e r' are two points such that p' *Vp, then pp, >c. See the Concluding Remarks at the end of this note. We say that the circles r and r, form a Villarceau pair of circles. Notice that the relationship between r and r, is symmetric, a fact already seen because the right-hand sides of (7) are obtained from those of (6) by replacing c by -c. Proof of Theorem 1. In the cartesian, axes O'xyz of Fig. I we have.1, 1)P: x c + aoo 0, y- a sin0, z 0-3-

9 We now rotate O'xyz about O'x by the angle a obtaining O'x'y'z' (O'X' M O'x). From P': x' - a cos f, y' " a sin *, z' 0, we find in O'xyz the coordinates x X, pi: y - y' cos a - z' sin a Z - y sin a + Z' COs a. Using (3) these become x - a cos (12) P': y b sin z c sin In terms of (11) and (12) the equation (8) becomes (PP') = (c + a cos 8 - a cos *) + (a sin 6 - b sin *) + c sin2 which easily gives (13) (pp') 2 c 2 +2a(a + c(cos 8-cos *) -a cos e co - b sin 8 sin *). * Therefore the property (8) is equivalent to the equation (14) a + c(cos e - cos *) - a cos 6 cos * - b sin 6 sin = * This is the equation between 8 and f which we want to solve for *. Doing this directly would be awkward as it would introduce complicated expressions and an extraneous solution. However, it becomes easy if we express and establish at the same time the properties (9): The components of the vector PP' are, by (11) and (12), (15) i-' (a co - c - a cos 0, b sin * - a sin 8, c sin *), [ 4

10 while dp To m (-a sin 0, a coo 8, 0) de - (-a sin #, b cos *, c cos ). Using inner products, the orthogonality properties (16) P 1 ", 6' I d p easily reduce, using (3), to (17) c sin e - a cos * sin 8 + b sin # coo 8 0 and (18) c sin * + a cos e sin # - b coso sin e, respectively. If we rearrange (14) and (17) as linear functions of con # and sin # we find the quivalent equations (19) (c + a coo O)cos # + b sin 9 sin * = a + c cos 8 a sin 0 cos - b cos sin = c sin 6. Solving (19) for coo * and sin * we readily find that (20) cos a coo 6 + c nb sin e Sa + c co sn a + c coo 0 A t 4 Iand the solution (6) is established: We easily verify that the right sides of t (20) are the coordinates of a point on the unit circle. It is also immediate that (20) satisfy identically the third equation (18). Solving the equations (20) for con 0 and sin 0, we find the equations (7). Notice that we have /1 established (6), (7), (8), and (9). i -5-

11 Proof of (10). Let the value ** in (12) produce the point P*'. By (13) and (14) we are to show the following: If (21) ** * *, (0 4 0 < 2R), then (22) F(e,$*) - a + c(cos 8 - Cos *) -a cos e cos ** - b sin 6 sin * > 0. A proof follows from the following facts: 1. F(e,f*) is a first order trigonometric polynomial in ** which vanishes for ** = P, by (14). 2. By (9) ** U * must be a double zero of F(e, **) and therefore F(e, **) > 0 for * sufficiently close to #. Since such a polynomial may have in its period at most two zeros, (22) is established. Concluding Remarks. 1. From (8) and (9) we see that r' is a Villarceau circle of the torus T having r as a central circle, and also that r is a Villarceau circle of the torus T' having r' as central circle. 2. Let us denote by T(r,c) a torus enveloped by a sphere of radius c whose center describes the circle r. From Fig. 1 we already know that the common normal PP' of r and r', is of constant length with PP' - c. We divide PP' into two parts with PQ - CI, QP' - c;, c, and c; constants, c, + c'i =c, and consider the two tori (23) T(r,c ) and T(r',c). i- ' '.-

12 I claims The two tori (23) are tangent to each other along all the points of a closed curve y described by the point Q. Proof: PQ is a normal of T(r,cI), while QP' is a normal of T(r',c'). It follows that the tori (23) are both tangent to the plane perpendicular to PP' at the point Q. Let (24) T(rc and T- be two tori that are linked, like two consecutive rings of a chain we also assume these tori to have equal central circles of radii - a. The diameter of the "hole" of T is 2(a - cl). Since T' passes through this hole, we must have 2c' < 2(a c or a > cl. + c'. Corollary 1. We can so place the linked tori (24) that they are tangent to each other along a simple closed curve y which is not a circle. Proof: We set c - c 1 + c' and identify the tori (24) with those of (23), and place them in the position of Fig. 1. As already mentioned, the tori (24) are now tangent to each other along the curve y described by the point Q with PQ - ci, P'Q m c 1. Notice that if we let becomes a Villarceau circle of c I + c and c' + 0, then the torus T(r',c;) "+ T(r,c). Conversely, Corollary I imples Villarceau's property, or V-property of tori: If Q is a point of contact of the tori, then their normals QP ac 1 and QP' - c; fit together to give the constant shortest distance.i PP' - c between the circles r and r'. circle of the torus T(F,c). Therefore r' is a Villarceau -- i -i

13 A pair of linked tori T and T', made of wood or lucite, having equal central circles can be used to exhibit to the "man on the street" their tangency in an appropriate position. This seems easier than showing the intersection of the torus by a bi-tangent plane. For a different approach to the Villarceau circles see 0. Bottema's note [ 3. REFERENCES Bottema, Cirkels op een torus, Pythagoras, 19 (1979) Z. A. Nelzak, Invitation to Geometry, John Wiley & Sons, New York, Y. Villarceau, Comptes Rendus de l'acadbmie des Sciences, Paris, IJS/Jik '1.1. _

14 SECURITY CLASSIFICATION Of THIS PAGE (Whn. Date 5ailweQ REPORT DOCUMENTATION PAGE FORM 1. REPORT NUMBER 12. GOVT ACC ESS IION NO, S. RECIPIUNT'S CATALOG NUMBER A D-A qi S3 4. TITLE (an IubittleJ S. TYPE Of REPORT A PERIOD COVERED Summary Report - no specific A Direct Approach to the Villaz-ceau circles reporting period of a Torus 41. PERFORMING ORG. REPORT NUMBER 7. AUTt4OR(e) S1. CONT-RACT OR GRANT NUMBER~a) I. J. Schoenberg DAAG29-80-C PERFORMING ORGANIZATION NAME AND ADDRESS SO. PROGRAM ELEMENT. PROJECT, 'TASK AREA & WORK UNIT NUMBERS Mathematics Research Center, University of W'brk Unit Number Walnut Street Wisconsin Miscellaneous Topics Madison, Wisconsin It. CONTROLLING OPFICIENAME AND ADDRESS 11. REPORT DATE U. S. Army Research Office February 1984 P.O. Box IS. NUMBER OF PAGES Research Triangle Park, North Carolina I.MONITORING AGENCY MNICE a ADDRESS(f isfuomt froms Conuhrollitid Office) 1S. SECURITY CLASS. (of this report) 16. DISTRIBUTION STATEMENT (of $f14 Roupe") Approved for public release; distribution unlimited. 164L UNCLASSIFIED DCkLASSIFICATIONOOWNGRADING SC EDLE 17. DISTRIBUTION STATEMENT (of Mhe absraentre min bl Sock 20, It different hrow Report) IS. SUPPLEMENTARY NOTES IS. KEY WORDS (Continue an reveree side it necessary d Idenifr by bweek nuumbor) The torus, Helicoidal motion of a circle. 20. ABSTRACT (Continue on rovae side if nocoom& nmd Identify by block nmber) Let T(r,c) denote a torus enveloped by a sphere of radius c whose center describes a circle r' of radius a, (c < a). we call r the central circle of the torus T(r,c). DD AO S17 EDITIONOFINV618BSLT I JA75 F 473 1NOV SISOSSOETEUNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Stn. wue &,emd)

15 20. Abstract (cont.) Let T(r,c 1 ) and T(F',cl) denote two tori satisfying the following conditions: 1. The central circles r and r' have equal radii, both - a. 2. The two tari T(r,c I ) and T(r',c;) are linked, like the two consecutive elements of a chain. This requires that a > C 1 + c. It is shown that the two tori T(r,c 1 ) and T(w',c;) can be placed in such a position that they are tangent to each other along a closed curve Y (which is not a circle) without any gaps between the tori. The simple closed curve Y is not a circle. It is also shown that this property is equivalent to the circles on the torus discovered by the astronomer Yvon Villarceau in The paper is to appear in the Dutch periodical "Simon Stevin'. / t.

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